In the generalized connectivity problem, we are given an edge-weighted graph G = (V, E) and a collection D = {(S 1 , T 1 ), . . . , (S k , T k )} of distinct demands; each demand (S i , T i ) is a pair of disjoint vertex subsets. We say that a subgraph F ⊆ G connects a demand (S i , T i ) when it contains a path with one endpoint in S i and the other in T i . The goal is to identify a minimum weight subgraph that connects all demands in D. Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some well-studied problems such as Steiner forest, non-metric facility location, tree multicast, and group Steiner tree. Finding a non-trivial approximation ratio for generalized connectivity was left as an open problem. Our starting point is the first polylogarithmic approximation for generalized connectivity, attaining a performance guarantee of O(log 2 n log 2 k). Here n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O(log 3 n log 2 k) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the directed Steiner network problem, we obtain an O(k 1/2+ǫ ) approximation, which improves on the currently best performance guarantee ofÕ(k 2/3 ) due to Charikar et al. (SODA '98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a polylogarithmic approximation; this result improves on the previously known ratio which can be Ω(n) in the worst case.
We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke (Proc. STACS '08, pages 669-680) by devising a deterministic approach whose performance guarantee is 4.91 + ε. In addition, we study preemptive online algorithms, a sub-class of one-pass algorithms where we are only allowed to maintain a feasible matching in memory at any point in time. All known results prior to Zelke's belong to this sub-class. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges which is not necessarily a feasible matching.
Assortment planning of substitutable products is a major operational issue that arises in many industries such as retailing, airlines, and consumer electronics. We consider a single-period joint assortment and inventory planning problem under dynamic substitution with stochastic demands, and provide complexity and algorithmic results as well as insightful structural characterizations of near-optimal solutions for important variants of the problem. First, we show that the assortment planning problem is NP-hard even for a very simple consumer choice model, where each consumer is willing to buy only two products. In fact, we show that the problem is hard to approximate within a factor better than 1 − 1/e. Second, we show that for several interesting and practical customer choice models, one can devise a polynomial-time approximation scheme (PTAS), i.e., the problem can be solved efficiently to within any level of accuracy. To the best of our knowledge, this is the first efficient algorithm with provably near-optimal performance guarantees for assortment planning problems under dynamic substitution. Quite surprisingly, the algorithm we propose stocks only a constant number of different product types; this constant depends only on the desired accuracy level. This provides an important managerial insight that assortments with a relatively small number of product types can obtain almost all of the potential revenue. Furthermore, we show that our algorithm can be easily adapted for more general choice models, and we present numerical experiments to show that it performs significantly better than other known approaches.
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